Abstract
For a Poisson algebra A, by exploring its relation with Lie–Rinehart algebras, we prove a Poincaré–Birkhoff–Witt theorem for its universal enveloping algebra Ae. Some general properties of the universal enveloping algebras of Poisson Hopf algebras are studied. Given a Poisson Hopf algebra B, we give the necessary and sufficient conditions for a Poisson polynomial algebra B[x;α,δ]p to be a Poisson Hopf algebra. We also prove a structure theorem for Be when B is a pointed Poisson Hopf algebra. Namely, Be is isomorphic to B#σH(B), the crossed product of B and H(B), where H(B) is the quotient Hopf algebra Be/BeB+.
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